Ultra high resolution gravity gradiometery technique

ABSTRACT

A method and system provide the ability to measure relative displacement and rotation. A first light is continuously shined from a first light source that is fixed on a first entity to a first two-dimensional (2D) plate fixed on a second entity. The first direction of propagation of the first light does not change relative to the first entity. A second light is shined from a second light source that is fixed on the first entity to a second 2D plate fixed on the second entity. A second direction of propagation of the second light does not change relative to the first entity and is different from the first direction of propagation. The displacement of the lights on plates is directly monitored to determine a 3D displacement vector that represents a relative displacement between the first entity and the second entity. Thus, a three dimensional gravity gradient tensor is constructed.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit under 35 U.S.C. Section 119(e) ofthe following commonly-assigned U.S. provisional patent application(s),which is/are incorporated by reference herein:

Provisional Application Ser. No. 61/579,573, filed on Dec. 22, 2011, byShervin Taghavi, entitled “Novel remote sensing technique based onmeasuring the relative displacement between two satellites,”;

Provisional Application Ser. No. 61/619,290, filed on Apr. 2, 2012, byShervin Taghavi, Massimo Tinto, Jakob J. VanZyl, Michael R. Hoffmann,and Christopher Boxe, entitled “Novel remote sensing technique based onmeasuring the relative displacement between two satellites,”; and

Provisional Application Ser. No. 61/665,793, filed on Jun. 28, 2012, byShervin Taghavi, Jakob J. VanZyl, Michael R. Hoffmann, and ChristopherBoxe, entitled “Novel Remote Sensing Technique Based on Measuring theRelative Displacement between Two Satellites,”.

This application is related to the following patent application, whichapplication is incorporated by reference herein:

Provisional Application Ser. No. 61/558,297, filed on Nov. 10, 2011, byShervin Taghavi, entitled “Novel technique for measuring with greatresolution the relative displacement and rotation between satellites,airplanes, etc. . . . ,”.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates generally to measuring the relativedisplacement between two entities (such as satellites, airplanes, etc.)or between any moving or static entities, wherein such displacement ismeasured in three (3) dimensions.

2. Description of the Related Art

(Note: This application references a number of different publications asindicated throughout the specification by reference numbers enclosed inbrackets, e.g., [x]. A list of these different publications orderedaccording to these reference numbers can be found below in the sectionentitled “References.” Each of these publications is incorporated byreference herein.)

The measurement and study of variations in the acceleration caused bygravity is called gravity gradiometry. This technique has been used toexamine subsurface geology to aid hydrocarbon (oil), water, mineralexploration, tunnel and bunker detection. The most frequently used andintuitive component is the vertical gravity gradient, G_(yy), whichrepresents the rate of change of vertical gravity (g_(y)) with height(y). As an example, a person walking past at a distance of two (2)meters would provide a gravity gradient signal approximately equivalentto 10⁻⁹ m/s² [3].

Recently, the Gravity Recovery and Climate Experiment (GRACE) (which wasa joint mission of NASA and the German Aerospace Center) and the GravityRecovery and Interior Laboratory (GRAIL) mission (which is part ofNASA's Discovery Program) have used a gravity gradiometry technique formapping the gravitational field of the Earth and the Moon, respectively.Both missions are very similar to each other; they are composed of twotwin satellites following each other in the same orbit, [1,2]. As thefirst satellite passes over a region of stronger gravity called agravity anomaly, it is pulled ahead of the trailing satellite. Thisincreases the separation distance between the two satellites. As thefirst satellite moves away from the anomaly, it decelerates; meanwhilethe second satellite approaches the anomaly, therefore it accelerates.The combination of these two phenomena induces a decrease in therelative distance between the two satellites. By constantly monitoringthe relative distance between the two satellites, scientists are able toconstruct a detailed map of Earth's and Moon's gravity.

One can notice that the major instruments of these techniques are theranging system called the Lunar Gravity Ranging System (LGRS) in thecase of GRAIL and High Accuracy Inter-satellite Ranging System (HAIRS)in the case of GRACE. The LGRS and HAIRS measure the relativedisplacement of the two satellites along the path way, which is almostthe orbit of the two satellites referred to as z. The ranging system,which is the heart of the current satellite gravity gradiometertechnique, is based on the Doppler Effect, which measures the phasechange of a wave travelling from one satellite to the other to determinethe change of the relative distance between the two satellites. However,such a ranging system only measures the distance along the axis ofpropagation of the E&M (electromagnetic) field, while not making anymeasurement on the two other perpendicular directions, thus being unableto construct a full tensor gradiometer. It is noteworthy to observe thatthe most frequently used and intuitive component of the gravity field isthe vertical gravity gradient, which for a circular orbit isperpendicular to the direction of propagation of the two satellites.Indeed, the orbit of the satellites is assumed to be a surface ofconstant gravity potential U and let r₁ and r₂ be positions on thesurface (i.e., with U₁ and U₂=U), then the component of g along the axisof the orbit is zero.

The g=−grad·U has no components along the axis of the orbit. Thisillustrates how important is it to measure the relative displacement ofthe two satellites in the plane horizontal to the axis separating thetwo satellites, which the currently technology based on the Dopplertechnique doesn't address.

Accordingly, what is needed is the ability to accurately (e.g., withouta significant impact due to noise) measure the relative displacementbetween two satellites in three (3) dimensions without relying on asignal's phase.

SUMMARY OF THE INVENTION

Embodiments of the present invention allow the measurement of therelative displacement between the two satellites in the two directionsthat the Doppler technique doesn't allow, thus the three-dimensionalrelative displacement vector, (ΔL_(x),ΔL_(y),ΔL_(z)), (a vitalobservable for reconstructing the components of the gravity field) canbe reconstructed. More specifically, embodiments of the inventionmeasure the relative displacement between the two satellites bymonitoring the displacement of the spot light created on one of the twosatellites from a light beam shined from the other satellite. Thistechnique is singular in the sense that it doesn't rely on the phase ofthe light, measuring directly the relative displacement of the twosatellites as opposed to a radar technique that measures the change ofthe phase and relates this change to the change of the distance. Bymonitoring the relative displacement between the two satellites, therelative displacement can be related to the relative accelerationbetween the two satellites using:

$\begin{matrix}{{{{\Delta\; a_{y}} = \frac{\left\lbrack {{\Delta\;{L\left( t_{1} \right)}_{y}} - {2\Delta\;{L\left( t_{2} \right)}_{y}} + {\Delta\;{L\left( t_{3} \right)}_{y}}} \right\rbrack}{2t_{d}^{2}}},{with}}{t_{2} = {t_{1} + t_{d}}}{and}{t_{3} = {t_{1} + {2t_{d}}}}} & \left( {{eq}.\mspace{14mu} 1} \right)\end{matrix}$where t_(d) is the detection rate (the time separation between twoconsecutives measurement). It may be noted that for measuring therelative displacement, one doesn't need to know the distance separationbetween the two entities and embodiments of the invention directlymeasure the relative displacement between the two; as opposed to theDoppler Effect, which deduces the relative displacement by measuring andcalculating the phase induced by a wave travelling from one satellite tothe other one and by monitoring the change in the phase of the wave atthe detection. By using drag-free accelerometers that providemeasurements to remove non-gravitational forces, one may deduce (fromthe previous equation) the variation of the gravity vector along the ydirection. This equation could be extended to the three spatialdirections.

The displacements that one is looking for are relatively small and atmost tens of μm. As an example, mountains induce a change in thegravitation force acceleration of several 10 μm/s⁻², [3]. GRAIL andGRACE are reporting a resolution of respectively 4 and 10 μm/s [1, 2],which means that they need a longer integration time to detect astructure of higher resolution while also requiring a longer baseline.Also the resolution limit of their ranging system doesn't allow them todetect fine human made underground structures. Distinguishably,embodiments of the present invention allow a relative displacementbetween the two satellites as small as few 10⁻¹⁸ m/s, which is twelveorders of magnitude better than what GRACE and GRAIL are capable of.This implies that one is able to detect the relative accelerationbetween the two satellites as small as few 10⁻¹⁸ m/s² or 10⁻¹⁷ m/s². Inorder to have an idea of its significance, let's assume that theexistence of an underground tunnel of infinite length of radius R=5 m,buried underground at a depth Y=200 m in a material made of concrete ofdensity ρ=2.4 g/cm³. The perturbation in the vertical component of thegravitational acceleration Δg_(y) is given by [4];

$\begin{matrix}{{\Delta\;{g_{y}\left( {10\mspace{20mu}\mu\; m\;\text{/}s^{2}} \right)}} = {0.2\;\left( \frac{R}{5\mspace{11mu} m} \right)^{2}\left( \frac{\left( {Y + h} \right)}{10\mspace{14mu} m} \right)^{- 1}\left( \frac{\rho}{2{gcm}^{3}} \right)}} & \left( {{eq}.\mspace{14mu} 2} \right)\end{matrix}$

For a pair of satellites flying at an altitude of h=500 km, thisrequires measuring the change in the relative distance with a precisionof 4.8×10⁻¹² m/s², which the ranging systems used by GRAIL or GRACEdon't provide. GRACE or GRAIL are looking for structures underground oron the surface for which the change in the gravity field induced by itis within their ranging system detection abilities, meaning a few μm/s².This change in the gravity field is commonly provided by a relativelyrough resolution's structure of few tens or hundreds of kilometers, thatis why the distance between their two satellites is within this range.

The precision that is required for detecting the example of the buriedunderground tunnel structure is 10⁶ times smaller than what GRAIL andGRACE can achieve in one second. In order to reach such resolution,GRAIL and GRACE would need to integrate over a period of 10¹² seconds,which is equivalent to more than thirty one thousand years ofintegration time. Embodiments of the present invention reach suchresolution by just integrating over 1 ns. Due to the high resolution ofdetection in embodiments of the invention, one can detect the change ofthe gravity field induced by structures of a few meters in diameter thatare buried deep underground. Thus, embodiments of the invention canafford to have a separation distance between the satellites as small asfew meters or even smaller.

Embodiments of the invention also provide a fast data rate acquisitiontime. GRAIL and GRACE are interested in long integration time forpurposes that include; monitoring long term trend change on the planet'sgravitational potential and the need to collect many data points todecrease the statistical error restrained by their ranging systemsresolution of few μm/s. The ranging systems of GRAIL and GRACE are basedon the dual one-way ranging system that has the advantage of eliminatingonboard Ultra Stable Oscillator noise for detection time longer than thetime it takes for the light to travel from one satellite to the otherone, which is typically in the range of ms. That also means that GRACEand GRAIL don't intend to collect data faster than a few ms. Embodimentsof the invention don't have to obey such limitations; and a detectionresolution (achieved by embodiments of the invention) of ˜10⁻¹⁸ m/s or0.2 pm/ns is good enough to not require the long integration times usedby GRAIL or GRACE.

The limitation in embodiments of the invention is a practical limit,meaning detecting data as fast as the technology used onboard allows it,basically continuously. It is not a fundamental limit as in GRAIL andGRACE. The fundamental limitation (of embodiments of the presentinvention) is the speed at which data can be detected and is referred toas the photon counting regime. The photon counting regime is describedby the time separating two consecutive photons hitting the detector,which is much faster than the technical limit as opposed to GRAIL andGRACE where their limitation in how fast they can detect data is afundamental limit and not a technical one. In view of the above,embodiments of the invention enable a detection time t_(d) of 1 ns,which is technically easily feasible. If the two satellites are orbitingin low orbit of earth, typically few hundred kilometers, they have avelocity of few km/s. Such properties enable embodiments of theinvention to detect data on the ground every μm.

BRIEF DESCRIPTION OF THE DRAWINGS

Referring now to the drawings in which like reference numbers representcorresponding parts throughout:

FIGS. 1( a) and 1(b) illustrate the general concept of shining lightbetween two satellites and measuring a relative displacement inaccordance with one or more embodiments of the invention;

FIG. 2 illustrates a satellite detecting the (Y,Z) components of arelative displacement between the two satellites in accordance with oneor more embodiments of the invention;

FIG. 3A illustrates an initial case where no relative displacementoccurs in accordance with one or more embodiments of the invention;

FIGS. 3B and 3C depict cases where there is a relative displacement inaccordance with one or more embodiments of the invention;

FIG. 4 illustrates a schematic of a quad cell technique in accordancewith one or more embodiments of the invention;

FIG. 5 illustrates an ensemble of four lenses (FIG. 5A) or mirrors (FIG.5B), where each lens (FIG. 5A) or mirror (FIG. 5B) focuses the incominglight in a different PMT in accordance with one or more embodiments ofthe invention;

FIG. 6 illustrates a one-dimensional amplified scheme of where thelight-beam hits, depicted by a cone in accordance with one or moreembodiments of the invention;

FIG. 7 illustrates a simplified scheme of the rotation of one satelliterelative to the incoming light in accordance with one or moreembodiments of the invention;

FIG. 8 represents the displacement induced by the rotation of thedetecting satellite relative to the z axis separating the two satellitesin accordance with one or more embodiments of the invention;

FIG. 9 represents the square root of the power spectrum density of thenoise of the relative displacement between the two satellites versus theFourier frequency, which is the frequency at which data is taken inaccordance with one or more embodiments of the invention;

FIG. 10 illustrates the logical flow for measuring a relativedisplacement and rotation in accordance with one or more embodiments ofthe invention; and

FIG. 11 is an exemplary hardware and software environment used toperform computations in accordance with one or more embodiments of theinvention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

In the following description, reference is made to the accompanyingdrawings which form a part hereof, and which is shown, by way ofillustration, several embodiments of the present invention. It isunderstood that other embodiments may be utilized and structural changesmay be made without departing from the scope of the present invention.

Qualitative Description

To understand embodiments of the invention, a qualitative description isuseful, followed by a the derivation of an analytical expression.

One may assume the existence of two satellites and it is desirable tomeasure their relative displacement by monitoring the displacement of abeam of light located on one of the satellites and shining to a platefixed to the other satellite. FIGS. 1( a) and 1(b) illustrate thegeneral concept of shining light between two satellites and measuring arelative displacement in accordance with one or more embodiments of theinvention. Assume that the light source is located on satellite 1 andshines 102 continuously on satellite 2. Therefore, the direction ofpropagation (Z) of the emitted light 102 never changes, relative tosatellite 1. Assume that the light source shines 102 on a plate (P₂)located in the (x,y)-plane and is fixed to satellite 2. FIG. 1( a)illustrates the initial setup of such a configuration. If the locationof the two satellites, relative to each other, does not change, then thelight-spot on satellite 2 remains at the same location on plate P₂. Itis assumed that, initially, without any relative displacement of the twosatellites, the light source shines 102 at the center O₂ of P₂. It isalso assumed that one satellite is displaced from the other satellite ina direction perpendicular to the X direction of propagation of thelight. If this distance, relative to satellite 2 is {right arrow over(ΔL)}, then the light-spot is displaced from O₂ by the same amount (asdepicted in FIG. 1( b)).

While one is able detect the relative displacement of the two satellitesnormal to the direction that the light propagates, one is not able todetect a relative displacement parallel to the direction where the lightpropagates, which is Z as shown in FIG. 1. This particular restrictioncan be surpassed of one considers using either the conventional DopplerShift technique or the scheme depicted in FIG. 2 to measure the relativedisplacement along the radial axis in the Z direction. The latter schemewill reveal the three-dimensional displacement vector {right arrow over(ΔL)} (ΔL_(x), ΔL_(y), ΔL_(z)) between the two satellites. One can thencollect at least two sets of independent data.

In the scheme of FIG. 2, a light source is emitted from satellite 1 thatpropagates in the Z direction and is monitored on a two-dimensionalplate located in the (X,Y) plane, and an additional light source locatedin satellite 1 that propagates in the (Y,Z) plane and is detected by atwo-dimensional plate located in the (Z,Y) plane. The latter scheme willreveal the three-dimensional displacement vector {right arrow over(ΔL)}(ΔL_(x), ΔL_(y), ΔL_(z)) between the two satellites. Accordingly,at least two sets of independent data may be collected. Thus, FIG. 2illustrates a satellite detecting the (Y,Z) components of a relativedisplacement between the two satellites in accordance with one or moreembodiments of the invention. As background for the concept ofdetermining displacement along a single axis, one may refer to [6] whichis incorporated by reference herein.

As well as monitoring the change in the relative distance between thetwo spacecrafts, it is also useful to recognize which spacecraft causesthe change. Here, one may define (1,2) and (2,1) as changes that occurrelative to satellite 1 and 2, respectively. For example, is it atrajectory change of satellite (1,2) that causes a relative displacementbetween the two satellites? This is done by using a two-way detectionscheme, where spacecraft 1 is lasing at spacecraft 2, butsimultaneously, spacecraft 2 is also lasing at spacecraft 1.

FIG. 3A illustrates an initial case where no relative displacementoccurs. FIGS. 3B and 3C depict cases where there is a relativedisplacement in accordance with one or more embodiments of theinvention. When there is no change in the relative displacement betweenthe two satellites, the light source from satellite (1,2) illuminates atthe origin of the plate located on satellite (2,1) as depicted in FIG.3A.

One may assume that at time t₀, satellite 1 moves and causes a change inthe separation distance between the two spacecraft as depicted in FIG.3B. One nano-second later, spacecraft 1 observes a displacement of thelight being detected on its photo-detector matrix, coming fromspacecraft 2. After the light travels from spacecraft 1 to spacecraft 2,spacecraft 2 will observe a displacement of the spot-light shining onits photo-detector matrix. If one considers that the distance betweenthe two spacecrafts is 10 m, and data is being acquired every ns, thenthe time it takes for the light to propagate from one spacecraft to theother is roughly 3.333 μs, corresponding to 3333 different data-sets,which corresponds to the acquisition of 3333 data-sets.

Similarly, as illustrated in FIG. 3C, if satellite 2 moves and satellite1 is stationary, at the next detection cycle, the emitted light fromsatellite 1 causes a separation distance with respect to satellite 2. Asimilar change is recorded by satellite 1 with a delay associated withthe duration taken for the light to travel from satellite 2 tosatellite 1. This exemplifies a two-way-scheme, where each spacecraftsends and receives light from the other spacecraft, which allows fordetermining which spacecraft causes a change in the separation distanceof the two spacecrafts—i.e., as long as the data is collected at leasttwice as fast the time it takes for the light to travel from onespacecraft to the other one.

Analytical Expression and General Notation

One may assume that each light source is secured and fixed to itsrespective satellites. The detection plane at satellite 2 may bearbitrarily chosen to be located in the (x,y)-plane, such that thez-axis is normal to the plane of detection. M_(t) ₀ , is denoted as thelocation on the plane where the center of the beam emitted fromsatellite 1 is projected, with a direction of propagation {right arrowover (k₁)}, at time t₀. One may assume that between t₀ and t₁, satellite2 displaces by {right arrow over (ΔL)}₂ from satellite 1, resulting in{right arrow over (M _(t) ₀ M)}_(t) ₁ =[ΔL _(2x)]{right arrow over (u_(x))}+[ΔL _(2y)]{right arrow over (u _(y))}.  (eq. 3)Therefore ΔL_(2x)=(M_(t) ₀ M_(t) ₁ ) and ΔL_(2y)=(M_(t) ₀ M_(t) ₁ ). Ifone takes into account the displacement {right arrow over (ΔL)}₁ inducedby the first satellite, which takes a time

$\frac{L}{c},$(L is the distance between the two satellites, and c is the speed oflight in vacuum) before reaching satellite 2, then,

$\begin{matrix}{{\overset{\longrightarrow}{M_{t_{0}}{M_{t_{0} + t}(t)}} = {{\left( {{\Delta\;{L_{2y}(t)}} - {\Delta\;{L_{1y}\left( {t - \frac{L}{C}} \right)}}} \right)\overset{\longrightarrow}{u_{y}}} + {\left( {{\Delta\;{L_{2y}(t)}} - {\Delta\;{L_{1y}\left( {t - \frac{L}{C}} \right)}}} \right)\overset{\longrightarrow}{u_{z}}}}},} & \left( {{eq}.\mspace{14mu} 4} \right)\end{matrix}$Likewise, the displacement of the beam on satellite 2 can be measured,which doubles the number of equations and, therefore, via the Fourier orLaplace transform, allows for the construction of the three-dimensionaldisplacements ΔL₂ and ΔL₁.Detection

This section describes two different modes of detection which aredepicted in FIGS. 4, 5A, and 5B. In this regard, incoming light may bedetected without using any mirror at the detection, or using an ensembleof mirrors, where each one of the mirrors focuses the incoming lightinto a different PMT (Photo Multiplier Tube).

FIG. 4 illustrates a schematic of a quad cell technique in accordancewith one or more embodiments of the invention. FIG. 5 illustrates anensemble of four lenses (FIG. 5A) or mirrors (FIG. 5B), wherein eachlens (FIG. 5A) or mirror (FIG. 5B) focuses the incoming light in adifferent PMT. This method of detection, similar to the Shack-Hartmannwavefront sensing technique, can be used to measure smaller relativedisplacements as the lenses produce focused light beams, collected bytheir respective PMTs, compared to the more dispersed light beam, whichinitially hits the lenses prior to being focused. Hence, using the quadcell technique with lenses or mirrors to focus the beam to its opticallimit of

${\sim\sim\frac{\lambda}{\pi}},$one can measure relative displacements as small as

$\frac{\lambda/\pi}{\sqrt{G \cdot N_{photon}}}$˜a few pico-meters per ns as in the specific example described herein.

Given a light beam's width, W, and signal size, SSSS=N _(photon) ·G,  (eq. 5)equals the number of photons N_(photon) detected multiplied by the gainG of the Photo Multiplier Tubes (PMTs). The quad cell technique of FIG.4 can be used to detect its displacement, ΔL, with a resolution of

${\frac{W}{\sqrt{SS}}\lbrack 5\rbrack},$which is equal to

$\frac{W}{\sqrt{N_{photon} \cdot G}}$per detection. If the detection rate is t_(d), then each second, t_(d)different data is detected, which leads to the detection resolution persecond to be:

$\begin{matrix}{{\Delta\;{L\left( {{meters}\text{/}{second}} \right)}} = {\frac{W({meter})}{\sqrt{N_{photon} \cdot G}} \cdot \sqrt{t_{d}({second})}}} & \left( {{eq}.\mspace{14mu} 6} \right)\end{matrix}$

In the quad cell technique, the detection panel is at least decomposedof four quadrants, each one corresponding to a different photo-detector.Let one denote i={1,2,3,4}, as the four quadrants, and (I_(t))_(t) ₀ ,as the light intensity (I), collected by a quadrant i at t₀ time [6]. Ifthe beam experiences a displacement, ΔL, between t₀ and t₁, then itsx-component, ΔL_(x), can be measured as:

$\begin{matrix}{{\Delta\; L_{x}} \simeq {{W \cdot \left( \frac{\left( {I_{2} + I_{4}} \right) - \left( {I_{1} + I_{3}} \right)}{I_{1} + I_{2} + I_{3} + I_{4}} \right)_{t = t_{1}}} - {W \cdot {\left( \frac{\left( {I_{2} + I_{4}} \right) - \left( {I_{1} + I_{3}} \right)}{I_{1} + I_{2} + I_{3} + I_{4}} \right)_{t = t_{0}}.}}}} & \left( {{eq}.\mspace{14mu} 7} \right)\end{matrix}$Likewise, its y-component, ΔL_(y), can be measured as:

$\begin{matrix}{{\Delta\; L_{y}} \simeq {{W \cdot \left( \frac{\left( {I_{1} + I_{2}} \right) - \left( {I_{3} + I_{4}} \right)}{I_{1} + I_{2} + I_{3} + I_{4}} \right)_{t = t_{1}}} - {W \cdot {\left( \frac{\left( {I_{1} + I_{2}} \right) - \left( {I_{3} + I_{4}} \right)}{I_{1} + I_{2} + I_{3} + I_{4}} \right)_{t = t_{0}}.}}}} & \left( {{eq}.\mspace{14mu} 8} \right)\end{matrix}$

In order to obtain the scheme shown in FIG. 4, one can either directlydetect the incoming light without using any mirrors at the detection orby using an ensemble of four (4) lenses (e.g., Lens 1 and Lens 2), whereeach lens focuses the incoming light into a different PMT (e.g., PMT 1and MPT 2) as illustrated in FIGS. 5A and 5B. FIGS. 5A and 5B illustratethat this method of detection can be used to measure smaller relativedisplacements as the lenses produce focused light beams, collected bytheir respective PMTs, compared to the more dispersed light beam, whichinitially hits the lenses prior to being focused. Hence, using the quadcell technique with lenses to focus the beam, relative displacements of˜a few pico-meters per detection [7] (meaning nano-second) can bemeasured.

A single lens (or mirror) solely would not allow for a relativedisplacement measurement of the incoming light as it focuses it to thesame point independently of the motion of the incoming plane-wave light;the mirror will always focus it at the same location, which is its focalplane. In other words, the focal plane of a lens or mirror is fixed andindependent of the proprieties of the incoming light. Focusing theincoming light with a single lens results in the loss of informationrelated to the displacement of the light. In order to overcome thisproblem embodiments of the invention utilize four (4) differentlenses/mirrors—each one focusing a quadrant of the incoming beam to itsrespective focal point, where a PMT is located at each focal plane. Asthe incoming beam moves, the relative light intensities detected by eachPMT change accordingly—thus, allowing for the reduction of the incominglight beam-width hitting the lenses, which in turn, allows that focusedbeam to reach its optical/fundamental size limit, which is

${{\sim\frac{\lambda}{\pi}} = {0.2\mspace{14mu}{\mu m}}},$if λ is 633 nm, [7]. The quad cell technique (applied to the 4 PMTs)then quantifies the relative displacement by measuring the relativechange in light intensities between respective PMTs, [5]. Given that Wis reduced to its fundamentally smallest optical size; this relativedisplacement is equal to that experienced by the light hitting the lensprior to being focused at each quadrant.Noise Analysis

Embodiments of the invention may utilize a technique similar GRACE andGRAIL to decouple non-Earth gravitational field forces from Earthgravitational field forces, both of which induce a relative displacementbetween the two satellites. Similar hardware, such as drag-freeaccelerometers and software can also be used in embodiments of thepresent invention. The sources of error in such a technique arequantitatively delineated in the following section. For example,embodiments of the invention enable a greater resolution of detection,which implies shorter integration times, and allows the detection ofgravity anomalies caused by relative smaller structures; this correlatesto integration times varying from a few nanoseconds to, at most,fractions of seconds. Described below is an examination of thesystematic errors that occur at timescales longer than integrationtimes, which may not be the case for GRAIL and GRACE. In addition, asdescribed herein, a two-way detection scheme of embodiments of theinvention may further help in selecting the useable data.

Pointing Stability

For a light half-width of 1 mm at a satellite separation-distance of 10m, a pointing stability of 5 milli-degree of the emitting light isrequired to not lose track of the light at the detection plane/panel. Tohave an idea, LISA (Laser Interferometer Space Antenna) reports apointing stability of 7 nano-rad [8], seven orders of magnitude betterthan what may be required. Amon et al. [9] deploys an opticalarchitecture, similar to embodiments of the invention, where theseparation between the two satellites is 3000 km.

Depending on the architecture of the satellites, it is expected that therange of the vibrations or the wobbling of the satellites that aremechanical motions will only be a few Hz to tens of Hz. These vibrationshappen at frequencies that are much smaller than embodiments theinvention's GHz rate of detection. Therefore, they can be seen assystematic and can be treated accordingly, given that due to highresolution of detection, one expects to not need long integration time.As seen earlier, a relative displacement between the two satellitesimply a displacement of the light beam at one satellite that is theopposite of the one collected at the other satellite. If satellitewobbling is the source of the recorded displacement, the previouscriteria is no longer satisfied. In other words, if the emittedsatellite shakes such that the detection beam hits a/point(s) outsidethe detection panel, the associated data is automatically discarded.This could then be used to eliminate the data that is thought to becaused by wobbling rather than the relative displacement between the twosatellites. The trend of the wobbling and vibration of each satellitecould be modeled (as a function of frequencies, angles, directions) andsimulated before flight. This model could be used to treat the wobblingas a systematic process that can be used to differentiate erroneous fromuseable/real data. Overall, sensors may be installed on the satellites,that monitor in real-time the physical variable that could lead to thewobbling of the satellite. For example, let one say that the change ofambient temperature or the radiation pressure induces a certain degreeof wobbling to the satellite. By having earlier learned and testedexperimentally how these changes induce wobbling and modeling them, itis possible to monitor, in real-time, the light-beam wobbling induceddisplacement, and separate such displacement from the one induced by therelative displacement of the two satellites. Thus, such data can beextracted from the wobbling noise.

Clock Noise

One of the advantages of embodiments of the invention, over the phasemeasurement technology, consists of not being limited by the UltraStable Oscillator (USO) noise in high frequency. Even in the case of aDual One-Way-Range measurement, which reduces the USO noise, acutoff-frequency for data acquisition exists where the USO noise can nolonger be reduced, thus deteriorating strongly the signal to noiseratio. Within this context, as long as the frequency at which data iscollected is less than

$\frac{C}{L},$then the USO noise cancels. This limitation is governed by the time ittakes for a wave to travel from one satellite to the other [10].Embodiments of the invention do not have this limitation since therelative displacement of the two satellites with respect to each otheris being measured continuously. This yields many advantages, such asmonitoring very fast processes that induce changes in the gravitypotential experienced by the two satellites; low-resolution dataproducts at the surface and subsurface can then be resolved.

Medium-Induced Noise

In the case of a phase detection system, such as Radar or Lidar, themedium in which the signal propagates can induce a phase error since itsindex of refraction can change due to a change in the concentration ofcharged particles, such as electrons. This particular error does notexist in embodiments of the invention, as the Doppler Shift technique isnot used. Instead, the relative displacement of the satellites caused bya change in the gravity field of a planetary body is measured. However,impurities in the medium can cause the scattering of the light. Changesin the scattering of the light over its respective time-scale mightinduce an error(s), which likely is much longer than the time separatingtwo consecutive detections. One should realize that since a differentialmeasurement between two different times is being performed, any sourceof error that is constant over this period will be canceled.

Effect of Non-Gravitational Forces

Non-gravitational forces, including atmospheric drag, solar radiationpressure, Earth radiation pressure, thrust, etc., can also induce arelative displacement of the two satellites [10]. Similar to GRACE,GRAIL and LISA, a drag-free accelerometer may be used at the center ofmass of each satellite, which only detects the total non-gravitationalforces. The accelerometer measurement can be used to identify thenon-gravitational force acting on the satellite trajectory and to removeits effect on the satellite trajectory estimation. If a similar dragfree accelerometer is utilized (e.g., similar to one proposed for use onLISA [8]), then the square-root of the power spectrum of theacceleration noise associated with it is

${3.10^{- 15} \cdot \frac{m}{s^{2} \cdot \sqrt{Hz}}},$which corresponds to a square-root of the power spectrum of thedisplacement of

${\frac{3.10^{- 15}f^{- 2}}{\left( {2\pi} \right)^{2}} \cdot \frac{m}{\sqrt{Hz}}},$where f is the Fourier frequency.

Throughout the following formulation, one may assume that the intensity,I(r, z), of the beam of light exhibits a Gaussian distribution. Manylasers operate in their fundamental traverse mode, TEM_(oo). Thisapproximates a Gaussian light beam in the plane perpendicular to thedirection of the propagating light. Then, the amplitude of the lightintensity of the electric field has a magnitude given by:

$\begin{matrix}{{{I\left( {r,z} \right)} = {I_{0} \cdot \left( \frac{w_{0}}{w(z)} \right)^{2} \cdot {\exp\left( \frac{{- 2}r^{2}}{{w(z)}^{2}} \right)}}},{where}} & \left( {{eq}.\mspace{14mu} 9} \right) \\{{{w(z)} = {w_{0} \cdot \sqrt{1 + \left( \frac{z}{z_{R}} \right)^{2}}}},} & \left( {{eq}.\mspace{14mu} 10} \right) \\{{z_{R} = \frac{\pi \cdot w_{0}^{2}}{\lambda}},} & \left( {{eq}.\mspace{14mu} 11} \right)\end{matrix}$r is the radial distance from the center-axis of the beam, z is theaxial distance from the beam's narrowest point (i.e., the “waist,”w(0)), and w(z), the waist-size), λ is the wavelength of the light beam,and z_(R) is the Rayleigh distance. The best resolution one can measurewith any displacement of a light beam of width w(z) with a signal sizeSs is [5]:

$\begin{matrix}{{{\Delta\; L} = \frac{w(z)}{\sqrt{Ss}}},} & \left( {{eq}.\mspace{14mu} 12} \right)\end{matrix}$which is independent of the nature of its distribution.

Skewness of the Data

One may assume that the incoming beam is not a perfect Gaussian beam butrather has some skew. Since a differential measurement is performed, anyexhibited skew in the distribution of the light beam does not affectmeasurements in embodiments of the invention.

Change in the Wavelength

A change in the wavelength doesn't displace the beam, but changes thewidth of the beam, thus changing the resolution of which thedisplacement of the beam can be detected; therefore,

$\begin{matrix}{{{\frac{\partial\left( {\Delta\; L} \right)}{\partial{w(z)}}{\mathbb{d}{w(z)}}} = \frac{\mathbb{d}{w(z)}}{\sqrt{Ss}}},} & \left( {{eq}.\mspace{14mu} 13} \right) \\{{{\mathbb{d}{w(z)}} = {\left( \frac{w_{0} \cdot z}{z_{R}} \right)^{2} \cdot \frac{1}{w(z)} \cdot \frac{\mathbb{d}\lambda}{\lambda}}},} & \left( {{eq}.\mspace{14mu} 14} \right) \\{{{\mathbb{d}\left( {\Delta\; L} \right)_{\lambda}} = {\frac{\mathbb{d}{w(z)}}{\sqrt{Ss}} = {\left( \frac{w_{0} \cdot z}{z_{R}} \right)^{2} \cdot \frac{1}{w(z)} \cdot \frac{1}{\sqrt{Ss}} \cdot \frac{\mathbb{d}\lambda}{\lambda}}}},} & \left( {{eq}.\mspace{14mu} 15} \right) \\{{{\mathbb{d}\left( {\Delta\; L} \right)_{\lambda}} = {\left( \frac{w_{0} \cdot z}{z_{R}{w(z)}} \right)^{2} \cdot \frac{\mathbb{d}\lambda}{\lambda}}},} & \left( {{eq}.\mspace{14mu} 16} \right)\end{matrix}$where Ss, the signal size of the light at the detection, is equated to

$\begin{matrix}{{Ss} = {\frac{I\left( {r,z} \right)}{hv} \cdot {t_{s}.{Then}}}} & \left( {{eq}.\mspace{14mu} 17} \right) \\{{{\mathbb{d}\left( {\Delta\; L} \right)_{\lambda}} = {\frac{\mathbb{d}\left( {w(z)} \right)_{\lambda}}{\sqrt{Ss}} = {\left( \frac{w_{0} \cdot z}{z_{R}} \right)^{2} \cdot \frac{1}{w(z)} \cdot \frac{\sqrt{h \cdot v}}{\sqrt{{I\left( {r,z} \right)} \cdot t_{s}}} \cdot \frac{\mathbb{d}\lambda}{\lambda}}}},} & \left( {{eq}.\mspace{14mu} 18} \right)\end{matrix}$where t_(s) is the time of the data acquisition, h the plank constant,and v is the frequency of the light.

Change in Intensity of the Light Beam

A change in the light beam intensity does not induce any displacement ofthe beam. However, it changes the resolution at which such adisplacement can be detected. For instance,

$\begin{matrix}{{\frac{\mathbb{d}\left( {\Delta\; L} \right)}{\mathbb{d}I} = {{{- \frac{1}{2}} \cdot \frac{w(z)}{\sqrt{Ss}}}\frac{\mathbb{d}{Ss}}{Ss}}}{since}} & \left( {{eq}.\mspace{14mu} 19} \right) \\{{{Ss} = {\frac{I\left( {r,z} \right)}{hv} \cdot t_{s}}},{yields}} & \left( {{eq}.\mspace{14mu} 20} \right) \\{\frac{\mathbb{d}{Ss}}{Ss} = {\frac{\mathbb{d}{I\left( {r,z} \right)}}{I\left( {r,z} \right)}.}} & \left( {{eq}.\mspace{14mu} 21} \right)\end{matrix}$

One then substitutes eq. 20 and 21 into eq. 19, which yields

$\begin{matrix}{\frac{\mathbb{d}\left( {\Delta\; L} \right)}{\mathbb{d}I} = {{- \frac{1}{2}}{\frac{\mathbb{d}{I\left( {r,z} \right)}}{I\left( {r,z} \right)}.}}} & \left( {{eq}.\mspace{14mu} 22} \right)\end{matrix}$

Given an integration time of a few milli-seconds, the laser intensitychange is about 2% (5). This yields:

$\begin{matrix}{{\left. \frac{\mathbb{d}\left( {\Delta\; L} \right)}{\mathbb{d}I} \right.\sim 1}{\%.}} & \left( {{eq}.\mspace{14mu} 23} \right)\end{matrix}$

This means that the change in the intensity of the laser source, whichhappens at very low Fourier-frequencies, induces a change of the shotnoise, at the detection of only 1%. Since this change is alreadyinsignificant at low Fourier-frequencies, this is even more negligibleat high Fourier-frequencies.

Inhomogeneity in the Photo-Multipliers

One may assume that all the photo-multiplier tubes are the same andtheoretically deliver the same output, I_(o), for the same input lightintensity, I. The PMTs do not all have the same dark current, I_(d), andthe same gain, G. Their respective light intensity output expression israther written as:I _(o) =I _(d) +GI.  (eq. 24)

The inhomogeneity has two components. The first one is the fluctuationof the dark current from one PMT to another. Since a differentialmeasurement is being made, between two different times, these termscancel—as long as they remain constant between two consecutivemeasurements. The second term is the variation of the gain of the PMTfrom one PMT to another one. The gain of each PMT can be calibratedprior to a measurement.

Rotation-Induced Displacement

An additional source of error that could interfere with the measurementis the rotation of the satellites. This is depicted by a cone, while thehitting plane is depicted by a line separating W_(L) and W_(R) asillustrated in FIG. 6. This causes a shift in the center of the beam aswell as changes in the Gaussian distribution, including its axis,widths, and center. FIG. 6 illustrates this phenomenon, where the lightpropagation is described by two lines representing a cone and the planeof the detection line. In other words, FIG. 6 illustrates aone-dimensional amplified scheme of where the light-beam hits, depictedby a cone. Initially, the center of the intersection between the coneand the line is M_(n), and the width of the intersection is|(W_(R))_(n)|. As the plane of detection rotates around the z-axis withan angle α, the new center intersection segment M_(n+1) differs,compared to M_(n), and its width changes (FIG. 7). Accordingly, FIG. 7illustrates a simplified scheme of the rotation of one satelliterelative to the incoming light. The beam is depicted by the cone, whilethe hitting plane is depicted by a line separating W_(L) and W_(R).

The equation of the two lines describing the cones is:

$\begin{matrix}{{Z = {{{\pm \frac{L_{z}}{W}} \cdot X} + L_{z}}},} & \left( {{eq}.\mspace{14mu} 25} \right)\end{matrix}$where L_(z) is the distance separating the two satellites and theequation of the detection line after the rotation isZ=αX.  (eq. 26)

The intersection points between the cone and the line are denoted(W_(L))_(n+1) and (W_(R))_(n+1), which satisfy eq. 25 and 26. The centerof these two points is called M_(n+1) and denotes the new center of thelight beam at the detection source. By using eq. 25 and 26, one candeduce that

$\begin{matrix}{{{M_{n + 1}M_{n}}} = {{L_{z} \cdot \alpha}{\frac{\sqrt{1 + \alpha^{2}}}{\alpha^{2} - \frac{L^{2}}{W^{2}}}.}}} & \left( {{eq}.\mspace{14mu} 27} \right)\end{matrix}$

Similarly, satellite rotation induces a change of the beam-width of|(W _(R))_(n+1)−(W _(R))_(n))−((W _(L))_(n+1)−(W _(L))_(n))|.  (eq. 28)

It is expected that the displacement caused by a rotation occurs ontimescales much longer than a nano-second detection time, and therefore,can be treated as a systematic component.

Numerical Application

One can assume that the distance separating the two satellites is 10 m,and a He/Ne laser is used, emitting at 632.9 nm. The half aperture-sizeof the spherical mirror collimating the light at the emission is w(0)=1mm; therefore, its width at the reception is w(z)=2.25 mm. One caneither detect the beam directly without changing its beam-width bydeploying a batch of Photomultiplier Tubes or using a setting similar toFIGS. 5A and 5B, which shrinks the beam-width (with the aid of at least4 lenses) to

${\sim\frac{\lambda}{\pi}} = {0.2\mspace{20mu}\mu\;{m.}}$If one assumes that 4 million photons hit the PMTs, at each timeframefor detection, with the gain G of 10⁶ for each PMT, the resolution ofmeasuring the relative displacement of the two satellites per detection(t_(d))) of 1 ns is:

$\begin{matrix}{{\Delta\; L} = {\frac{W}{\sqrt{{SS} = {G.N_{photon}}}} = {\frac{0.2\mspace{20mu}\mu\; m}{\sqrt{10^{6} \cdot 10^{6}}}\text{/}{ns}}}} \\{= {0.2\mspace{14mu}{pm}\text{/}{{ns}.}}}\end{matrix}$

Within 1 s of integration time, one collects 10⁹ data, therefore anintegration time of 1 s will reduce the shot noise error of themeasurement by a factor of √{square root over (t_(d)=10⁻⁹)} whichcorresponds to a displacement of

0.2 pm. √{square root over (10⁻⁹)}·s⁻¹ equals to 6.32×10⁻¹⁸ m/s.

This scheme requires a light intensity at the detection of 1.26 mW.Taking into account the attenuation factor of the light between theemissions to the detection of

${\left( \frac{w(z)}{w(0)} \right)^{2} = 5.0625},$yields an emission intensity of 6.35 mW. Table 1 and 2 (below) provide acomplete summary of all parameters associated with the design andnoise-related sources.

TABLE 1 Numerical value of various physical parameters. SatellitesDistance of separation between satellites 10 m Satellites speed on earth7.8 km s⁻¹ Satellite altitude 500 km Spatial resolution on earth 7.8 mmDetection time 1 ns Beam Beam width w(0) 1 mm Spherical mirror aperturesize at emission 2 mm Beam width at the second satellites w(z) 2.25 mmLenses size at detection 50 mm Beam width at the PMT's ~μm Rayleighdistance 4.96 m PMT'S PMT's size Few mm Detection time 1 ns Gain 1millions Number of photons detected per PMT 1 millions Total number ofphotons at detection 4 millions Laser Laser wavelength 632.9 nm Laserintensity ~2.11 mw Relative displacement detection resolution 0.28 pmper detection time Relative displacement detection resolution 6.32 ×10⁻¹⁸ m/s per second Detection time t_(d) 1 ns${{Light}'}s\mspace{14mu}{intensity}\mspace{14mu}{attenuation}\mspace{14mu}\left( \frac{w(z)}{w(0)} \right)^{2}$5.06 Intensity at detection ~1.26 mW Intensity at emission ~6.35 mW

TABLE 2 Noise sources and measurement resolution changes. DisplacementDisplacement induced Resolution change Laser intensity change None$\frac{{d({\Delta L})}_{I}}{\Delta L} = {\frac{dI}{I} < {2\%}}$ Laserwavelength change None$\frac{{d({\Delta L})}_{\lambda}}{\Delta L} = {0.8 \times 10^{- 4.5}\frac{d\;\lambda}{\lambda}}$PMT dark current None None PMT inhomogeneity None None if calibratedMirrors None None at first order if inhomogeneity calibrated Relativesatellite <1 μm per rotation Pico meter rotation Drag free accelerometer$\frac{3.10^{- 15}.}{\left( {2\pi\; f} \right)^{2}}\frac{m}{\sqrt{Hz}}$$\frac{3.10^{- 15}.}{\left( {2\pi\; f} \right)^{2}}\frac{m}{\sqrt{Hz}}$

FIG. 8 represents the displacement induced by the rotation of thedetecting satellite relative to the z axis separating the twosatellites. Crucial missing information is the timescale at which theserotations take place. One expects the rotations to happen in a timingrange larger than the detection rate utilized in embodiments of theinvention. For a large rotation, the techniques discussed above (e.g.,in the Pointing Stability section) may be used to differentiate the realdata from the one induced by the rotation of the satellites andtherefore treating the latter as systematics. One may also note that therequired relative pointing stability of the spacecraft in the case ofLISA where the satellites are meant to be 5 million kilometers apart, isδθ˜7 nrad/√{square root over (Hz)} [8], which is the square-root of thepower spectrum density of the angular fluctuation δθ which correspondsto 0.4 μdeg.

One feature of FIG. 8 is that for the range of the angle of interest,the induced displacement is a linear function of the rotation angle,which may facilitate the extraction of the rotation induceddisplacement's pattern from the collected data. In addition, one canactually fulfill the requirement to differentiate data by intentionallyalleviating many stringent requirements on the satellites by allowingthem to rotate in ranges that induce displacement of a few orders ofmagnitude larger that the range of displacement that is sought. Such anapproach enables one to differentiate the data being sought fromsystematics. Essentially any source of error that the frequencyassociated with is different than the expected short integration time.In addition, if the magnitude of its induced measured displacement isdifferent than the ones induced by the structures sought, it is possibleto extract it from the measurements.

With regard to laser-induced errors, one may observe that laserintensity and wavelength fluctuations induce changes in the resolutionof measuring a relative displacement that are quite negligible comparedto the shot noise of the measurement. One can also show that theintensity of the light beam for measurements require low power and is inthe range of the power of commonly/widely used lasers. Furthermore,embodiments of the invention do not require any specific stabilizationfeature for potential laser sources, as those already available on themarket may fulfill any necessary requirements.

For the LISA mission, the drag-free accelerometer meant to decouple thenon-gravitational force from the gravitational forceinduced-displacement is a function of the square-root of its noise powerspectrum of the drag-free accelerometer—i.e.,

${3 \times 10^{- 15}\text{/}\left( {2\pi\; f} \right)^{2}{\frac{meters}{\sqrt{Hz}}.}},$[8]. FIG. 9 represents the square root of the power spectrum density ofthe noise of the relative displacement between the two satellites versusthe Fourier frequency, which is the frequency at which data is taken.For instance, it is known that the corresponding shot-noise of eachmeasurement is 0.2 pm per detection time. For each measurement, one hasto account for the noise contribution of the displacement caused by bothsatellites, which can then be summed till a threshold of 0.28 pm. If onedefines ΔL(f)=(ΔL)²t_(d), as the power spectrum density of the relativedisplacement, such a definition leads to

${{\Delta\;{L(f)}} = {7.84 \times 10^{- 35}\frac{{meters}^{2}}{Hz}}},$corresponding to a square root of the power spectrum of

${\sqrt{7.84 \times 10^{- 35}}\frac{meters}{\sqrt{Hz}}} = {8.8 \times 10^{- 18}{\frac{meters}{\sqrt{Hz}}.}}$If the separation distance between the two satellites is extended to 100km, that would lead to a relative square root of the power spectrum of

$\frac{\sqrt{\Delta\;{L(f)}}}{L} = {8.8 \times 10^{- 23}\text{/}{\sqrt{Hz}.}}$At this level of sensitivity, this scheme could be used forgravitational wave detection as well. FIG. 9 depicts the square-rootpower spectrum of the relative displacement between the two satellites,including the contribution of the drag free accelerometer to the totalnoise.Logical Flow

FIG. 10 illustrates the logical flow for measuring a relativedisplacement and rotation in accordance with one or more embodiments ofthe invention.

At step 1002, a first light is continuously shined from a first lightsource that is fixed on a first entity to a first 2D plate fixed on asecond entity. A first direction of propagation of the first light doesnot change relative to the first entity.

At step 1004, a second light is continuously shined from a second lightsource that is fixed on the first entity to a second 2D plate fixed onthe second entity. A direction of propagation of the second light doesnot change relative to the first entity. Further, the first direction ofpropagation and the second direction of propagation are different.

At step 1008, the displacement of the first light on the first plate andthe second light on the second plate is directly monitored to determinea 3D displacement vector that represents a relative displacement (inthree dimensions) between the first entity and the second entity.

In embodiments of the invention, the first and second entity may both besatellites. Further third and fourth light sources may be shined fromthe second entity to the first entity (in a manner similar to that ofthe first light and the second light) and monitored. Based on thedisplacement monitoring from all of the lights, a determination can bemade regarding which entity caused the relative displacement (i.e.,which entity moved relative to the other entity). Such a determinationmay be made based on a delay associated with a duration taken for thelights to travel from one entity to another entity. In this regard, thedisplacement monitoring is conducted faster than a time it takes for thelight to travel from one entity to another. The rate of collecting datafor the displacement vector is determined by a photon counting regime.As long as the photons arriving at a plate that is receiving the lightdo not reach/exceed the photon counting regime, data can be collected.As used herein, the photon counting regime comprises a time separatingtwo consecutive photons hitting a detector.

In embodiments of the invention, relative displacement as small as

$\frac{W(z)}{\sqrt{SS}}$is measured, where W is the width of the first light in a z-directionafter being reduced to its optical limit of

$\frac{\lambda}{\pi}$and SS=G·N_(photon) is a signal-size being detected at a detection whereG comprises a gain of an optical detector and N_(photon) is a number ofphotons. In a similar manner, the relative displacement in the x and/ory directions can be determined (e.g., the relative displacement ismeasured in an x-direction and/or a y-direction after being reduced toits optical limit).

In addition to the above, the 3D displacement vectors may be used toquantify a gravity potential of any entity (referred to as a thirdentity). In this regard, the gravity potential can be utilized tospatially and temporally measure the physical parameters of any entitythat is inducing change in the gravity potential. Such third entitiesmay include a planetary body, an object, or a gravitational wave. As anexample, such objects may be a macroscopic object and/or a microscopicobject such as for example a molecule, a cell, a gravitational wave, orany wave that induces a change in the gravity.

One can also use the same technique described above to determine arelative rotation between one of the entities relative to the otherentity. Instead of using a centroid error algorithm to monitor therelative displacement of the center of the beam and therefore deduce therelative displacement of the detecting entity relative to the emittingentity, the change of the width W of the beam at the detection may bemonitored to deduce the relative rotation between two entities.

Computer Utilization

The computations/determining/etc. described herein may be conductedusing one or more devices within or exterior to the entities (e.g.,satellites, planes, etc.) described herein. For example, a computerinside of a satellite that is configured with various processors andprocessing capabilities may be configured to perform the variousoperations described herein.

FIG. 11 is an exemplary hardware and software environment 1100 used toperform such computations in accordance with one or more embodiments ofthe invention. The hardware and software environment includes a computer1102 and may include peripherals. Computer 1102 may be a user/clientcomputer, server computer, or may be a database computer. The computer1102 comprises a general purpose hardware processor 1104A and/or aspecial purpose hardware processor 1104B (hereinafter alternativelycollectively referred to as processor 1104) and a memory 1106, such asrandom access memory (RAM). The computer 1102 may be coupled to, and/orintegrated with, other devices, including input/output (I/O) devicessuch as a keyboard 1114, a cursor control device 1116 (e.g., a mouse, apointing device, pen and tablet, touch screen, multi-touch device, etc.)and a printer 1128. In one or more embodiments, computer 1102 may becoupled to, or may comprise, a portable or media viewing/listeningdevice 1132 (e.g., an MP3 player, iPod™, Nook™, portable digital videoplayer, cellular device, personal digital assistant, etc.). In yetanother embodiment, the computer 1102 may comprise a multi-touch device,mobile phone, gaming system, internet enabled television, television settop box, or other internet enabled device executing on various platformsand operating systems.

In one embodiment, the computer 1102 operates by the general purposeprocessor 1104A performing instructions defined by the computer program1110 under control of an operating system 1108. The computer program1110 and/or the operating system 1108 may be stored in the memory 1106and may interface with the user and/or other devices to accept input andcommands and, based on such input and commands and the instructionsdefined by the computer program 1110 and operating system 1108, toprovide output and results.

Output/results may be presented on the display 1122 or provided toanother device for presentation or further processing or action. In oneembodiment, the display 1122 comprises a liquid crystal display (LCD)having a plurality of separately addressable liquid crystals.Alternatively, the display 1122 may comprise a light emitting diode(LED) display having clusters of red, green and blue diodes driventogether to form full-color pixels. Each liquid crystal or pixel of thedisplay 1122 changes to an opaque or translucent state to form a part ofthe image on the display in response to the data or informationgenerated by the processor 1104 from the application of the instructionsof the computer program 1110 and/or operating system 1108 to the inputand commands. The image may be provided through a graphical userinterface (GUI) module 1118. Although the GUI module 1118 is depicted asa separate module, the instructions performing the GUI functions can beresident or distributed in the operating system 1108, the computerprogram 1110, or implemented with special purpose memory and processors.

In one or more embodiments, the display 1122 is integrated with/into thecomputer 1102 and comprises a multi-touch device having a touch sensingsurface (e.g., track pod or touch screen) with the ability to recognizethe presence of two or more points of contact with the surface. Examplesof multi-touch devices include mobile devices (e.g., iPhone™, Nexus S™,Droid™ devices, etc.), tablet computers (e.g., iPad™, HP Touchpad™),portable/handheld game/music/video player/console devices (e.g., iPodTouch™, MP3 players, Nintendo 3DS™, PlayStation Portable™, etc.), touchtables, and walls (e.g., where an image is projected through acrylicand/or glass, and the image is then backlit with LEDs).

Some or all of the operations performed by the computer 1102 accordingto the computer program 1110 instructions may be implemented in aspecial purpose processor 1104B. In this embodiment, the some or all ofthe computer program 1110 instructions may be implemented via firmwareinstructions stored in a read only memory (ROM), a programmable readonly memory (PROM) or flash memory within the special purpose processor1104B or in memory 1106. The special purpose processor 1104B may also behardwired through circuit design to perform some or all of theoperations to implement the present invention. Further, the specialpurpose processor 1104B may be a hybrid processor, which includesdedicated circuitry for performing a subset of functions, and othercircuits for performing more general functions such as responding tocomputer program 1110 instructions. In one embodiment, the specialpurpose processor 1104B is an application specific integrated circuit(ASIC).

The computer 1102 may also implement a compiler 1112 that allows anapplication or computer program 1110 written in a programming languagesuch as COBOL, Pascal, C++, FORTRAN, or other language to be translatedinto processor 1104 readable code. Alternatively, the compiler 1112 maybe an interpreter that executes instructions/source code directly,translates source code into an intermediate representation that isexecuted, or that executes stored precompiled code. Such source code maybe written in a variety of programming languages such as Java™, Perl™,Basic™, etc. After completion, the application or computer program 1110accesses and manipulates data accepted from I/O devices and stored inthe memory 1106 of the computer 1102 using the relationships and logicthat were generated using the compiler 1112.

The computer 1102 also optionally comprises an external communicationdevice such as a modem, satellite link, Ethernet card, or other devicefor accepting input from, and providing output to, other computers 1102.

In one embodiment, instructions implementing the operating system 1108,the computer program 1110, and the compiler 1112 are tangibly embodiedin a non-transient computer-readable medium, e.g., data storage device1120, which could include one or more fixed or removable data storagedevices, such as a zip drive, floppy disc drive 1124, hard drive, CD-ROMdrive, tape drive, etc. Further, the operating system 1108 and thecomputer program 1110 are comprised of computer program 1110instructions which, when accessed, read and executed by the computer1102, cause the computer 1102 to perform the steps necessary toimplement and/or use the present invention or to load the program ofinstructions into a memory 1106, thus creating a special purpose datastructure causing the computer 1102 to operate as a specially programmedcomputer executing the method steps described herein. Computer program1110 and/or operating instructions may also be tangibly embodied inmemory 1106 and/or data communications devices 1130, thereby making acomputer program product or article of manufacture according to theinvention. As such, the terms “article of manufacture,” “program storagedevice,” and “computer program product,” as used herein, are intended toencompass a computer program accessible from any computer readabledevice or media.

Of course, those skilled in the art will recognize that any combinationof the above components, or any number of different components,peripherals, and other devices, may be used with the computer 1102.

CONCLUSION

This concludes the description of the preferred embodiment of theinvention. Measurements of the relative displacement of satellites viaonboard Doppler tracking are of great interest for inferring theinterior structure of planetary bodies from the observed gravity fieldsusing the gravity gradiometry technique. The fundamental limitations ofmissions that use this technique and rely on two satellites, such asNASA's GRACE, GRAIL, and the GRACE Follow-On missions, are constrainedby their Doppler technique ranging system limitations, which can't solvethe relative displacement between the two satellites better than 4 μm/s[1,2]; and also intrinsically to any Radar or Lidar technologies, theirmeasurements are only made only along the direction of propagation ofthe field. If z is the direction of propagation of the field along theaxis separating the two satellites, and a relative displacement in the(x,y)-plane occurs, then the Doppler technique will not detect it.

Embodiments of the invention overcome this limitation via a newtechnique referred to as the “Shervin Taghavi Larigani GravityGradiometry” (STLGG) technique, which directly measures the relativedisplacement in the transversal plane of the propagation of the field.Within this context, each satellite shines a laser beam to the othersatellite, where differential intensity measurements are performed bymultiple onboard photomultipliers. By properly combining thesedifferential intensity measurements, it may be shown that it is possibleto reconstruct the three-dimensional relative displacement vector,(ΔL_(x),ΔL_(y),ΔL_(z)), a vital observable for reconstructing thecomponents of the gravity field. Embodiments of the invention allow acontinuous data acquisition time as opposed the Dual One-Way Dopplertracking technique used by GRAIL and GRACE, which doesn't provide theability to take data faster than the time it takes light to travel fromone satellite to the other satellite, in the order of ms. Thesensitivity enhancement of embodiments of the invention, which measuresthe relative displacement of the two satellites at a resolution of6.32×10⁻¹⁸ m/s, twelve orders of magnitude better than the rangingsystems used by GRAIL and GRACE, results in improved temporal andspatial resolution of the data and allows a separation distance betweenthe two satellites of only a few meters.

Accordingly, a new remote sensing technique is introduced that is basedon measuring the relative displacement of two satellites. This isprimarily based on the fact that the density of an object, such asplanetary body, induces changes in its respective gravity field.

Embodiments of the invention derive a new technique that measures therelative displacement of a satellite directly. This technique allowsfor: (1) the measurement of the rate range (or relative displacement)between the two satellites, which does not rely on a USO, as opposed tophase detection (e.g., the Doppler technique); (2) a faster dataacquisition rate as the embodiments are not limited by light travelingtime between the satellites in order to suppress the USO noises; (3) themeasurement of the three-dimensional component of the relativedisplacement, which allows for more accurate determination of thespatial composition of the relative displacement; (4) a determination ofthe higher-harmonic components of a body's gravity potential; (5) animproved ground data resolution that could reach 11 orders of magnitude;(6) a more accurate localization of the source of changes in the gravityfield by sourcing which of the satellites that may contribute to causingthose changes; (7) a detection of high velocity processes and non-radarabsorbing materials in a planetary atmosphere; and (8) an increase inthe S/N ratio.

In view of the above, embodiments of the invention measure the relativedisplacement of a satellite directly and are both applicable tomeasuring the gravity field and gravitational wave. Accordingly,embodiments of the invention allow for resolving high resolutionstructure(s)/object(s) and processes, relevant for earth and planetarydiscovery, industrial applications (e.g., increases in the efficiencyfor air traffic control, better alignment of objects in space, etc.),and for a variety of military applications (e.g., detection of non-radarabsorbing materials, high velocity atmospheric objects (missiles,airplanes, etc.,) submarines, underground man-made structure, andmissile sites. In this regard, embodiments of the invention areuniversally-applicable to military and defense and industrialactivities, mining, locating subsurface oil reservoirs, locatingunderground contamination sources, and the retrieval of relevantatmospheric and surface/subsurface parameters.

The foregoing description of the preferred embodiment of the inventionhas been presented for the purposes of illustration and description. Itis not intended to be exhaustive or to limit the invention to theprecise form disclosed. Many modifications and variations are possiblein light of the above teaching. It is intended that the scope of theinvention be limited not by this detailed description, but rather by theclaims appended hereto.

If one extends the length of separation between two entities to forexample to 100 km, then a relative displacement measurement of

${\left. \frac{\Delta\; L}{L} \right.\sim 10^{- 23}} \cdot s^{- 1}$is reached, which is equivalent to a relative square root of a powerspectrum of

$\frac{8.8 \times 10^{- 23}}{\sqrt{Hz}}.$This can be used for the detection of gravitational wave.

REFERENCES

-   [1] Antreasian, P. G., et. al., Navigation of the Twin GRAIL    Spacecraft into Science Formation at the Moon. 23^(rd) International    Symposium on Space Flight Dynamics, Pasadena Convention Center    October 29-Nov. 2, 2012,-   [2] Byron Taple, Frank Flechtner, Michael M. Watkins, et. al,    Gravity Recovery and Climate Experiment: Key GRACE Facts. The Earth    Observing System Project Science Office, NASA,    http://eospso.gsfc.nasa.gov/eos_homepage/missionprofiles/docs/GRACE.pdf.-   [3] http://www.gradiometry.com/gradiometry-   [4] Cornwall, J., Despain, A., Eardley, D., Garwin, R., Hammer, D.,    Jeanloz, R., Katz, J., Rothaus, O., Ruderman, M., Schwitters, R.,    Treiman, S., & Vesecky, J. Characterization of Underground    Facilities. The MITRE Corporation, JASON Program Office, 1-68, 1999.-   [5] Sandrine Thomas, Optimized centroid computing in a    Shack-Hartmann sensor, Cerro Tololo Inter-American Observatory    archive, 2004.-   [6] Claire Max. Wavefront Sensing. Lecture 7, Astro 289C, UCSC, Oct.    13, 2011-   [7] A. E. Siegman, Lasers, University Science Books, 1986-   [8] LISA An international project in the field of Fundamental    Physics in Space, Pre-Phase A Report Second Edition July 1998.    http://lisa.gsfc.nasa.gov/Documentanon/ppa2.08.pdf-   [9] Amon, S., Rotman, S. R., & Kopeika, N. S. Performance    limitations of free-space optical communication satellite networks    due to vibrations: direct detection digital mode. Opt. Eng. Vol    36(11), 3148-3157, 1997.-   [10] Kim, J. In Simulation Study of a Low-Low Satellite-to-Satellite    Tracking Mission. Ph.D. Thesis, University of Texas at Austin, 2000.

What is claimed is:
 1. A method for measuring a relative displacementand rotation, comprising: continuously shining a first light from afirst light source that is fixed on a first entity to a firsttwo-dimensional (2D) plate fixed on a second entity, wherein a firstdirection of propagation of the first light does not change relative tothe first entity, and wherein the first 2D plate comprises a first quadcell; continuously shining a second light from a second light sourcethat is fixed on the first entity to a second 2D plate fixed on thesecond entity, wherein a second direction of propagation of the secondlight does not change relative to the first entity, and wherein thefirst direction of propagation is different from the second direction ofpropagation, and wherein the second 2D plate comprises a second quadcell; and monitoring displacement of the first light on the first plateand the second light on the second plate to directly determine athree-dimensional (3D) displacement vector that represents a relativedisplacement in three dimensions between the first entity and the secondentity, wherein the monitoring displacement utilizes a quad celltechnique to determine the relative displacement with a resolution of$\frac{W}{\sqrt{SS}},$ wherein W represents a width of a light beam ofthe first light and the second light, and SS represents a signal size ofthe first light and the second light.
 2. The method of claim 1, whereinthe first entity and the second entity are both satellites.
 3. Themethod of claim 1, further comprising: continuously shining a thirdlight from a third light source that is fixed on the second entity to athird 2D plate fixed on the first entity, wherein a third direction ofpropagation of the third light does not change relative to the secondentity; continuously shining a fourth light from a fourth light sourcethat is fixed on the second entity to a fourth 2D plate fixed on thefirst entity, wherein a fourth direction of propagation of the fourthlight does not change relative to the second entity, and wherein thethird direction of propagation is different from the fourth direction ofpropagation; monitoring displacement of the third light on the thirdplate and the fourth light on the fourth plate; and determining whichentity causes the relative displacement based on the displacementmonitoring from the first light, second light, third light, and fourthlight.
 4. The method of claim 3, wherein: the determining is based on adelay associated with a duration taken for the third light and thefourth light to travel from the second entity to the first entity; andthe displacement monitoring is conducted faster than a time it takes forthe third light and the fourth light to travel from the second entity tothe first entity.
 5. The method of claim 3, wherein: a rate ofcollecting data for the displacement vector is determined by a timeseparating two consecutive photons hitting a detector; and as long as aphoton arriving at the third plate do not reach the time, data iscollected.
 6. The method of claim 1, wherein: relative displacement assmall $\frac{W(z)}{\sqrt{SS}}$ as is measured, where W is the width ofthe first light in a z-direction after being reduced to its opticallimit of $\frac{\lambda}{\pi}$ and SS=G·N_(photon) is a signal-sizebeing detected at a detection where G comprises a gain of an opticaldetector and N_(photon) is a number of photons.
 7. The method of claim1, further comprising: utilizing the 3D displacement vector to quantifya gravity potential of a third entity.
 8. The method of claim 7, furthercomprising: utilizing the gravity potential to spatially and temporallymeasure physical parameters of the third entity that is inducing changein the gravity potential.
 9. The method of claim 1, wherein the 3Ddisplacement vector comprises a relative rotation between the firstentity and the second entity.
 10. An system for measuring a relativedisplacement and rotation comprising: (a) a first light source, wherein:(i) the first light source is fixed on a first entity; (ii) the firstlight source continuously shines a first light to a firsttwo-dimensional (2D) plate fixed on a second entity; (iii) the first 2Dplate comprises a first quad cell; and (iv) a first direction ofpropagation of the first light does not change relative to the firstentity; (b) a second light source, wherein: (i) the second light sourceis fixed on the first entity; (ii) the second light source continuouslyshines a second light to a second 2D plate fixed on the second entity;(iii) the second 2D plate comprises a second quad cell; (iv) a seconddirection of propagation of the second light does not change relative tothe first entity; and (v) the first direction of propagation isdifferent from the second direction of propagation; and (c) the secondentity is configured to directly monitor displacement of the first lighton the first plate and the second light on the second plate to determinea three-dimensional (3D) displacement vector that represents a relativedisplacement in three dimensions between the first entity and the secondentity, wherein the second entity directly monitors displacementutilizing a quad cell technique to determine the relative displacementwith a resolution of $\frac{W(z)}{\sqrt{SS}}$ wherein W represents awidth of a light beam of the first light and the second light, and SSrepresents a signal size of the first light and the second light. 11.The system of claim 10, wherein the first entity and the second entityare both satellites.
 12. The system of claim 10, further comprising: (d)a third light source, wherein: (i) the third light source is fixed onthe second entity; (ii) the third light source continuously shines athird light to a third 2D plate fixed on the first entity; and (iii) athird direction of propagation of the third light does not changerelative to the second entity; (e) a fourth light source, wherein: (i)the fourth light source is fixed on the second entity; (ii) the fourthlight source continuously shines a fourth light to a fourth 2D platefixed on the first entity; (iii) a fourth direction of propagation ofthe fourth light does not change relative to the second entity; and (iv)the third direction of propagation is different from the fourthdirection of propagation; (f) the first entity is configured to monitordisplacement of the third light on the third plate and the fourth lighton the fourth plate; and wherein: a determination regarding which entitycauses the relative displacement is based on the displacement monitoringfrom the first light, second light, third light, and fourth light. 13.The system of claim 12, wherein: the determination is based on a delayassociated with a duration taken for the third light and the fourthlight to travel from the second entity to the first entity; and thedisplacement monitoring is conducted faster than a time it takes for thethird light and the fourth light to travel from the second entity to thefirst entity.
 14. The system of claim 12, wherein: a rate of collectingdata for the displacement vector is determined by a time separating twoconsecutive photons hitting a detector; and as long as a photonsarriving at the third plate do not reach the time, data is collected.15. The system of claim 10, wherein: relative displacement as small as$\frac{\lambda}{\pi},$ is measured, where W is the width of the firstlight in a z-direction after being reduced to its optical limit of$\frac{w}{\sqrt{ss}},$ and SS=G·N_(photon) is a signal-size beingdetected at a detection where G is the gain of the optical detector andN_(photon) is the number of photons.
 16. The system of claim 10, furthercomprising: a quantifying entity that is configured to utilize the 3Ddisplacement vector to quantify a gravity potential of a third entity.17. The system of claim 16, wherein: the gravity potential is utilizedto spatially and temporally measure physical parameters of the thirdentity that is inducing a change in the gravity potential.
 18. Thesystem of claim 10, wherein the 3D displacement vector comprises arelative rotation between the first entity and the second entity.